# Algebraic Geometry Iv Linear Algebraic Groups Invariant by A.N. Parshin

By A.N. Parshin

Two contributions on heavily comparable topics: the idea of linear algebraic teams and invariant conception, by means of famous specialists within the fields. The booklet can be very priceless as a reference and study consultant to graduate scholars and researchers in arithmetic and theoretical physics.

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**Extra info for Algebraic Geometry Iv Linear Algebraic Groups Invariant Theory**

**Example text**

Since π ◦ σ is a morphism with rationally connected ﬁbres, we get c(Σ+ , P+ ) = 0. This example can be easily generalized to linear projections of Fano complete intersections V ⊂ PM of index 2 or higher. 2. For a mobile linear system Σ on a variety X deﬁne the virtual threshold of canonical adjunction by the formula cvirt (Σ) = inf {c(Σ , X )}, X →X where the inﬁmum is taken over all birational morphisms X → X, X is a smooth projective model of C(X), Σ is the strict transform of the system Σ on X .

This claim can be generalized in the following way [HasTsch]: for any points xi = Fti on non-singular ﬁbres Fti = π 1 (ti ) and any set of li -jets of non-singular curves transversal to the ﬁbres Fti at these points, there exists a section s : P1 → V with these jets at the points xi , (l1 , . . , lk ) ∈ Zk+ is an arbitrary set of non-negative integers. In other words, one can prescribe a section of tangent vectors (transversal to ﬁbres) and more generally jets of ﬁnite order at given points. 2 is a natural generalization of the classical Tsen theorem [Shaf, Kol96] on the existence of a section of the projection π : V → P1 , where V ⊂ Pm ×P1 is a hypersurface of bidegree (d, N ), d ≤ m.

A4 ) | t ∈ C} has at least a double tangency with V at the point x if and only if q1 (a∗ ) = q2 (a∗ ) = 0. Interpreting (a∗ ) as homogeneous coordinates on the projectivized tangent space P(Tx P4 ) ∼ = P3 , so that the plane {q1 = 0} ⊂ P3 is P(Tx V ), we see that the lines we are looking for are parametrized by the conic Q(x) = {q1 = q2 = 0} ⊂ P(Tx V ). We obtain a well-deﬁned map px : Q(x) → V . It is not too diﬃcult to construct explicit examples of quartics containing a rational surface S ⊂ V (which plays the same part as the rational curve C for the cubic hypersurface above).